Golden Triangle: What is It?

Date: 09/19/1999 at 16:27:01
From: Robin Byerly
Subject: Golden Triangle
I'm doing a project for math where I have to answer questions using
the Internet. I have gone to many different Web sites trying to figure
out what the Golden Triangle is, but I cannot find it. It has to do
with math but so far all my results have something to do with
bicycles. Please help me find out what the Golden Triangle is. Thank
you.
Robin Byerly

Date: 09/20/1999 at 05:34:49
From: Doctor Floor
Subject: Re: Golden Triangle
Hi, Robin,
Thanks for your question.
Suppose we have a triangle ABC, such that <A = <B = 72 degrees
(< means angle) and <C = 36 degrees. Such a triangle is known as the
Golden Triangle.
Let D be the point on BC, such that AD is the angle bisector of <A.
Then triangle ABD is again a Golden Triangle.
When we let lengths AB = AD = CD = x and BD = 1, then we find:
AB : BD = BC : AB
x : 1 = (x+1) : x
This can be rewritten to:
x^2 = x + 1
x^2 -x - 1 = 0
The two solutions for x are x = 1/2 +/- sqrt(5)/2. Since AB > BD, in
this case we must have x = 1/2 + sqrt(5)/2.
The number 1/2 + sqrt(5)/2 is known as the Golden Ratio, or Golden
Mean. So BC : AB is this famous ratio; that's why this triangle is
called a Golden Triangle.
For more about the Golden Ratio, see our FAQ:
Golden Ratio, Fibonacci Sequence
http://mathforum.org/dr.math/faq/faq.golden.ratio.html
As an example of the appearance of Golden Triangles: the outside
triangles of a pentagram are Golden Triangles.
When we attach to AC and BC two triangles that are congruent to
triangle ACD, we find a regular pentagon.
I hope this helped. If you need more, just write us back.
Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/